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TemperatureCylindrical.jl
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# Define a function to form the energy equation left hand side and right hand side
# In cylindrical coordinates!!
# x -> r
# y -> z
function assemble_energy_equation_cylindrical(grid::CartesianGrid,rho_c::Matrix{Float64},Cp_c::Matrix{Float64},kThermal::Matrix{Float64},H::Matrix{Float64},Tlast::Matrix{Float64},dt::Float64,bctype,bcval)
# Assemble the left hand side for the energy equation
# Inputs:
# grid - this is the CartesianGrid
# rho_c - density at the cell centers
# Cp_c - heat capacity at the cell centers
# kThermal- thermal conductivity at the basic nodes
# H - volumetric rate of internal heating at the cell centers
# Tlast - the previous-timestep temperature.
# dt - the time step
# bctype - a vector containing flags for the boundary condition at the [left, right, top, bottom]
# 1 = fixed temperature
# 2 = fixed dT/dn
# bcval - a vector containing the temperature or dT/dn values at the [left, right, top, bottom]
# Returns:
# L,R, the left hand side and right hand side of the energy equation
bcleft = bctype[1]#-1 # -1 = constant normal temperature gradient, 1 = constant temp
bcright = bctype[2]#-1 #
bctop = bctype[3]#1
bcbottom = bctype[4]#1
# bcval should contain temperature or dT/dx values for left,right,top,bottom
N = grid.nx*grid.ny
row = zeros(Int64,5*N);
col = zeros(Int64,5*N);
val = zeros(Float64, 5*N);
R = zeros(Float64,N,1);
k = 1;
# j-1 j j j+1 j+1
# i-1 |----kC----|----vy----|
# | | |
# i kA T kB T |
# | | |
# i |----kD--(i,j)--vy----|
# | | |
# i+1 vx c vx c vx
# | | |
# i+1 |----vy----|----vy----|
for j in 1:grid.nx
dxc = j>1 ? grid.x[j] - grid.x[j-1] : grid.x[j+1] - grid.x[j]
dxp = grid.xc[j+1] - grid.xc[j]
dxm = j>1 ? grid.xc[j]-grid.xc[j-1] : grid.xc[j+1] - grid.xc[j]
for i in 1:grid.ny
dyc = i>1 ? grid.y[i] - grid.y[i-1] : grid.y[i+1] - grid.y[i]
dyp = grid.yc[i+1] - grid.yc[i]
dym = i>1 ? grid.yc[i]-grid.yc[i-1] : grid.yc[i+1] - grid.yc[i]
this_row = node_index(i,j,grid.ny);
if i==1 # ghost nodes along top
row[k] = this_row
col[k] = this_row
val[k] = 1.0 #1.0/2.0
k+=1
row[k] = this_row
col[k] = node_index(i+1,j,grid.ny)
val[k] = bctop #bctop/2.0
k+=1
R[this_row] = bctop == 1 ? 2.0*bcval[3] : -dyp*bcval[3]
elseif j==1 # ghost nodes along left side.
row[k] = this_row
col[k] = this_row
val[k] = 1.0/2.0
k+=1
row[k] = this_row
col[k] = node_index(i,j+1,grid.ny)
val[k] = bcleft/2.0
k+=1
R[this_row] = bcval[1]
else
# kA, kB, kC, kD
kA = 0.5*(kThermal[i-1,j-1] + kThermal[i,j-1])
kB = 0.5*(kThermal[i,j] + kThermal[i-1,j])
kC = 0.5*(kThermal[i-1,j-1] + kThermal[i-1,j])
kD = 0.5*(kThermal[i,j-1] + kThermal[i,j])
#rho_c = 0.25*(rho[i-1,j-1] + rho[i,j-1] + rho[i-1,j] + rho[i,j])
#Cp_c = 0.25*(Cp[i-1,j-1] + Cp[i,j-1] + Cp[i-1,j] + Cp[i,j])
# diagonal entry
row[k] = this_row;
col[k] = this_row;
val[k] = (rho_c[i,j]*Cp_c[i,j])/dt + kB*grid.x[j]/grid.xc[j]/dxp/dxc + kA*grid.x[j-1]/grid.xc[j]/dxm/dxc + kD/dyp/dyc + kC/dym/dyc;
k+=1
# right
row[k] = this_row;
col[k] = j==grid.nx ? node_index(i,j,grid.ny) : node_index(i,j+1,grid.ny);
val[k] = j==grid.nx ? bcright*kB*grid.x[j]/grid.xc[j]/dxp/dxc : -kB/dxp/dxc*grid.x[j]/grid.xc[j];
k+=1
# left
row[k] = this_row;
col[k] = node_index(i,j-1,grid.ny);
val[k] = -kA*grid.x[j-1]/grid.xc[j]/dxm/dxc;
k+=1
# down (+y)
row[k] = this_row;
col[k] = i==grid.ny ? node_index(i,j,grid.ny) : node_index(i+1,j,grid.ny);
val[k] = i==grid.ny ? bcbottom*kD/dyp/dyc : -kD/dyp/dyc;
k+=1
# up (-y)
row[k] = this_row;
col[k] = node_index(i-1,j,grid.ny);
val[k] = -kC/dym/dyc;
k+=1
R[this_row] = Tlast[i,j]*rho_c[i,j]*Cp_c[i,j]/dt + H[i,j];
if j==grid.nx
R[this_row] += 2*bcval[2]*bcright*kB*grid.x[j]/grid.xc[j]/dxp/dxc
end
if i==grid.ny
R[this_row] += 2*bcval[4]*bcbottom*kD/dyp/dyc
end
end
end
end
row = @views row[1:k-1]
col = @views col[1:k-1]
val = @views val[1:k-1]
L = sparse(row,col,val)
return L,R
end
function subgrid_temperature_relaxation_center!(markers::Markers,grid::CartesianGrid,Tlast::Matrix,Cp,kThermal,dt::Float64)
# Perform the sub-grid scale temperature diffusion operation
# Inputs:
# markers - the markers
# grid - the grid
# Tlast - the previous timestep temperature solution at the cell centers
# Cp (scalar) heat capacity
# kthermal (scalar) thermal diffusivity
# dt - the time step
# Returns:
# a matrix whose values are the change in temperature at the cell centers
dsubgrid = 1.0; # subgrid temperature diffusivity
dT_subgrid_m = Array{Float64,2}(undef,1,markers.nmark)
# compuate the nodal temperature on the markers.
cell_center_to_markers!(markers,grid,Tlast,dT_subgrid_m)
# compute the subgrid temperature changes on the markers
rho = markers.scalarFields["rho"]
T = markers.scalarFields["T"]
Threads.@threads for i in 1:markers.nmark
dx2 = (grid.x[markers.cell[1,i]+1] - grid.x[markers.cell[1,i]])^2
dy2 = (grid.y[markers.cell[2,i]+1] - grid.y[markers.cell[2,i]])^2
tdiff = markers.scalars[rho,i]*Cp/kThermal / (2/dx2 + 2/dy2)
dT_subgrid_m[i] = (dT_subgrid_m[i]-markers.scalars[T,i])*( 1.0 - exp(-dsubgrid*dt/tdiff) )
end
# interpolate subgrid temperature changes back onto basic nodes.
markers.scalars[T,1:markers.nmark] += dT_subgrid_m[1,:]
# zero out nodal values for any cells without markers (nan values)
dTm, = marker_to_stag(markers,grid,dT_subgrid_m,"center")
dTm[isnan.(dTm)] .= 0.0
return dTm
end
function compute_boundary_heat_flow(grid::CartesianGrid,Tc::Matrix{Float64},kThermal::Matrix{Float64})
if size(Tc,1) != grid.ny+1 || size(Tc,2) != grid.nx + 1
error("temperature array needs to contain ghost values")
end
qsurf = (Tc[2,:] - Tc[1,:]) ./ (grid.xc[2]-grid.xc[1])
Qsurf = 0.0
for j in 2:grid.nx
qsurf[j] *= 0.5*(kThermal[1,j]+kThermal[2,j])
Qsurf += qsurf[j] * 2*pi*grid.xc[j]*(grid.x[j]-grid.x[j-1])
end
qbtm = (Tc[2,:] - Tc[1,:]) ./ (grid.xc[2]-grid.xc[1])
Qbtm = 0.0
for j in 2:grid.nx
qbtm[j] *= 0.5*(kThermal[grid.ny+1,j]+kThermal[grid.ny,j])
Qbtm += qbtm[j] * 2*pi*grid.xc[j]*(grid.x[j]-grid.x[j-1])
end
println("Surface heat flow, ",Qsurf," Bottom: ",Qbtm)
end